MODELING THE STORY OF FUNCTIONS

Standards for Mathematical PracticeCommon Core

Presenter: Ellen Falkefalk@northsalemschools.org

www.mathizaverb.comNorth Salem , NY

Mathematical Practice

From the Common Core Document under Mathematics: Standards for Mathematical Practice p 5 4. Model with mathematics.

“Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life,society, and the workplace. “

Context

“They routinely interpret their mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has notserved its purpose.”From the Standards of Mathematical practice #4

It’s Not Just About Content

Development of the pathways into instructional programs will require careful attention to modeling and the mathematical practices.

Assessments based on these pathways should reflect both the content and mathematical practices standards.”

Rules of Engagement

FOCUS

MATH WITH MATH

M

ATH

I

N

CO

NTEX

T

LOSS of :Width, Motivation, Applications

Loss of: DepthEfficiencyElegance

Representational Fluency

The Lesh translation model suggests that elementary mathematical ideas can be represented in five different modes: manipulatives, pictures, real-life contexts, verbal symbols, and written symbols. It emphasizes that translations within and between various modes of representation make ideas meaningful for students.

Performance Tasks

•Designed to reveal a learner's understanding of a problem/task and her/his mathematical approach to it.

•Can be a problem or a project, performance.

•It can be an individual, group or class-wide exercise.

•A good performance task usually has eight characteristics (outlined by Steve Leinwand and Grant Wiggins and printed in the NCTM Mathematics Assessment book).

•Good tasks are: essential, authentic, rich, engaging, active, feasible, equitable and open.

Project Based Learning

Investigations and meaningful tasks.

Construct knowledge through inquiry.

Culminates in a realistic hands –on project.

5 Es Instructional Model.

Linear Model -Discrete

Cutting String Simple, cheap! Use about 5 pieces of string

per person Domain, Range , independent variable,

dependent variable, constant rate of change, arithmetic sequence, linear function, linear regression, modeling.

Notation- How you want students to communicate their mathematics.

Problem solving How do we keep track of the data? How do we look for relationships?

Cut the String- no cuts

1 cut

2 cuts

3 cuts

Number of cuts n

Pieces of String

0 1

1 3

2 5

3 7

4 9

2

2

2

2

How to communicate the model

And the function is….S(n)=2n +1,

Recursively speaking..S1 = S0 +2

S2 = S1 +2

S3 = S2+2

S n+1= S n +2, S0 =1

From Regents Prep

Is there an issue with the subscripts when following this formula compared to the work on the previous slide?

More Models

Linear What’s Your Measure-continuous Crickets-continuous

Quadratic Leap Frog- wooden manipulaitve

NLVM Java applet ( Peg Puzzle) Empire State-continuous

Exponential Bust-A_Move ( NLVM java applet Tower of

Hanoi) The number “e”

LeAp FrOg

Create your data table.

Examine the common difference.

What do you notice? Pattern?

Take the common difference of the differences. What do you notice?

LeAp FrOgNumber

of Pairs n

Number of Moves

M1 3

2 8

3 15

4 24

5 33

5

7

9

11

2

2

2

Quadratic!!! Recall Calculus?

Tower Of Hanoi

Tower of HanoiNumber of disks

n

Number of Moves

M1 1

2 3

3 7

4 15

5 31

2

4

8

16

2

8

4

Exponential!!! Recall Calculus?

Summary Points

Start with discrete models.

Let them think! Let them LINK.

Free form modeling- let the student construct the solution in their own way then…

Formalize their communication.

Be positive problem solvers.

Summary Questions

What is meant by mathematical modeling?

How do we construct meaningful tasks?

Where do I start?

How many do we need?

Real, relevant, reliable, reusable.

Videos Dan Meyer-math class needs a makeover. RSA Animate-Ken Robinson

Hans Rosling : Population Growth over 200 years.

David McCandless turns complex data sets (like worldwide military spending, media buzz, into beautiful, simple diagrams that tease out unseen patterns and connections.

Taylor Mali- just because

Thanks!

Sources

Rational Number Project Lesh, R., Cramer, K., Doerr, H., Post, T., Zawojewski, J., (2003)

Using a translation model for curriculum development and classroom instruction. In Lesh, R., Doerr, H. (Eds.) Beyond Constructivism. Models and Modeling Perspectives on Mathematics Problem Solving, Learning, and Teaching. Lawrence Erlbaum Associates, Mahwah, New Jersey. Cramer, K., (2001) 

Using Models to Build Middle-Grade Students' Understanding of Functions. Mathematics Teaching in the Middle School. 6 (5), 310-318.mber Project

Davis, Robert, developer. Madison Project Independent Exploration Materials. Danbury Conn.: Math Media, 1966.